Distribution of dealer hands

I am interested in knowing the likelihood of various dealer hands given that the dealer draws to 12-16 with any number of cards. I am happy to disregard the effect of card removal for this calculation. Assume that it's a S17 game.

I have calculated the following:

If dealer has 16, then he has 1/13 chance of making either 17, 18, 19, 20 or 21 and he has a 8/13 chance of busting. I will call this p(17/16) ie. likelihood of drawing to 17 off a 16 = 7.69%
If dealer has 15, then again he has a 1/13 chance of making any of 17-21 plus he also has a 1/13 chance to draw to 16. At 16 he would then again have a 1/13 chance of getting 17-21 etc.

Thus for stiff hand of 15 I would say probability of getting 17 is 1/13 + 1/13x1/13 i.e. P(17/15) = 7.69% + 0.59% = 8.28%

Following on, likelihood of getting a 17 of a 14 stiff is:
1/13 (i.e. drawing a 3)
1/13 of getting a 2 and thus having a shot at p(17/16)
1/13 of getting a A and thus having a shot at p(17/15)
Thus this is equal to 7.69% + 1/13x7.69% + 1/13x8.28% = 8.92%

Using this approach I calculated P(17/13) to be 9.61% and P(17/12) to be 10.35%.

The numbers are all the same regardless whether we are looking for the dealer to draw to 17, 18, 19, 20 or 21.

This also helped me to determine that likelihood of the dealer to bust off 12, 13, 14, 15 and 16 are 48.27%, 51.96%, 55.39%, 58.58% and 61.54% respectively.

I'd like to know whether my calculations seem correct, or whetehr I have made an error somewhere?

You need to distinguish between hard and soft hands.

MangoJ said:

You need to distinguish between hard and soft hands.

Click to expand...

Thanks for pointing this out Mango! Though I was only meaning to analyse hard hands in this exercise.

a follow-up on this. i would also like to consider the likely hands when the dealer is on a hard 11.

Using my approach above, I think my answer would be:
1/13 + 1/13x (P17/16) + 1/13 x P(17/15) + .... + 1/13 x P(17/12)
= 11.14%

This would be the case for finishing hands of 17-20

For the likelihood of finishing on 21 it would be
4/13 + 1/13x (P17/16) + 1/13 x P(17/15) + .... + 1/13 x P(17/12)
=33.63%

Does that sound correct?

You are basically on the right way.

Thus for stiff hand of 15 I would say probability of getting 17 is 1/13 + 1/13x1/13 i.e. P(17/15) = 7.69% + 0.59% = 8.28%

Click to expand...

This will work, if you neglect the softness of an Ace. But that is not the game of Blackjack.

Drawing from 15 to 17 has multiple ways:
H15+2
H15+A+A
S15+A+A
S15+10+2
...
S15+7+5
S15+7+4+A

(you get the idea)

I'll give you a short spreadsheet line I wrote a few months ago, where you can calculate the probabilties for a S17 game with infinite deck, without peeking, and ignoring blackjack (i.e blackjack is counted as Soft 21): View attachment 7425

Instructions: All cells will be zero at first. To get probabilities, overwrite a single cell (i.e for "Hard 8") with the sure-value of 1.0.
All other cells give you the probability of holding this hand in the traverse of drawing. Red cells are the standing cells, so those cells give you the probability that the dealer stands on a Hard 19, given he had a Hard 8.

You should play with the file, if you are interested.

Easy problems are:
- calculate the pre-deal probability (without knowing the upcard)
- alter the file for a hit17 dealer
- incorporate a blackjack probability

Advanced problems:
- alter the file where the dealer has peeked at his hole card, and doesn't have blackjack
- Specify an (infinite deck) card distribution, to incorporate Spanish 21.

Expert problems:
- extend the file for card numbers (i.e. "3-card Hard 16"), and calculate the probability for a 6-card Charlie and 5-card 21

MangoJ said:

You are basically on the right way.



This will work, if you neglect the softness of an Ace. But that is not the game of Blackjack.

Drawing from 15 to 17 has multiple ways:
H15+2
H15+A+A
S15+A+A
S15+10+2
...
S15+7+5
S15+7+4+A

(you get the idea)

I'll give you a short spreadsheet line I wrote a few months ago, where you can calculate the probabilties for a S17 game with infinite deck, without peeking, and ignoring blackjack (i.e blackjack is counted as Soft 21): View attachment 7425

Instructions: All cells will be zero at first. To get probabilities, overwrite a single cell (i.e for "Hard 8") with the sure-value of 1.0.
All other cells give you the probability of holding this hand in the traverse of drawing. Red cells are the standing cells, so those cells give you the probability that the dealer stands on a Hard 19, given he had a Hard 8.

You should play with the file, if you are interested.

Easy problems are:
- calculate the pre-deal probability (without knowing the upcard)
- alter the file for a hit17 dealer
- incorporate a blackjack probability

Advanced problems:
- alter the file where the dealer has peeked at his hole card, and doesn't have blackjack
- Specify an (infinite deck) card distribution, to incorporate Spanish 21.

Expert problems:
- extend the file for card numbers (i.e. "3-card Hard 16"), and calculate the probability for a 6-card Charlie and 5-card 21

Click to expand...

I found this to be an interesting idea. I went ahead and added one more row for Hard 0 in order to compute overall dealer probabilities within the same context. In order to maintain valid data I found that undo can be clicked before setting a single cell's probability to 1 for a new input.

k_c said:

In order to maintain valid data I found that undo can be clicked before setting a single cell's probability to 1 for a new input.

Click to expand...

That will work too. I pretty much prefer to keep this column for reference and just duplicate the column. Then one can have different columns for different scenarios (i.e. different upcards)

hey MangoJJ, wantd to say thanks for that file u posted. forgot to say that earlier. i did have good look at it., many thanks again!!

MangoJ said:

Expert problems:
- extend the file for card numbers (i.e. "3-card Hard 16"), and calculate the probability for a 6-card Charlie and 5-card 21

Click to expand...

I just tackled the expert problem for nothing more than a hobby, with a simple modification of the spreadsheet file. It now also addresses a natural as a "2-card Soft 21".